
where \(u^*\)  is the decreasing rearrangement of \(u\) , see (REF ) below. Inequality (REF ) was proved by Bandle [1]} for strong solutions, by Vázquez [2]} and Mossino-Rakotoson [3]} for weak solutions. The elliptic version of (REF ) is known as Talenti's comparison [4]}, [5]}. The proof of (REF ) is mainly based on the isoperimetric inequality and symmetrization techniques. Talent and Bandle's comparisons play an important role in mathematical studies, since there provide sharp estimates for solutions to elliptic and parabolic equations, Faber-Krahn type inequality for eigenvalues [4]}, [7]}, and the bounds on the exit times of Brownian motion[8]}, [9]}.
Bandle's comparison (REF ) was generalized to nonlinear parabolic equations with Dirichlet boundary condition (see for instance [10]}, [11]} and references therein), and to parabolic equations with Neumann boundary condition [12]} as well. We also refer the reader to the excellent books [13]}, [14]} for related topics.
